Kato's inequality when $Δu$ is a measure

We extend the classical Kato's inequality in order to allow functions $u \in L^1_\mathrm{loc}$ such that $Δu$ is a Radon measure. This inequality has been applied by Brezis, Marcus, and Ponce to study the existence of solutions of the nonlinear equation $- Δu + g(u) = μ$, where $μ$ is a measure and $g : \mathbb{R} \to \mathbb{R}$ is an increasing continuous function.

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